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Chords of echidna

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Below are listed the dyadic chords of 11-limit echidna temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering by 540/539 are labeled swetismic, by 896/891 pentacircle, and by 176/175 valinorsmic.

The normal mapping for echidna is ech = [<2 1 9 2 12|, <0 3 -6 5 -7|]. From this we may derive a val v = ech[1] + 100 ech[2] = <2 301 -591 502 -688| which we may use to sort and normalize the chords of echidna. Under "Chord" is listed the chord, normalized to start from zero, in the mapping by v, and the next column gives a transversal which tempers to the chord by echidnic tempering. Finally, the Graham complexity is listed in the last column.

Echidna has MOS of size 6, 8, 14, 22, 36, 58 and 80. It may be seen that a modest number of chords are available to 14 notes, but that 22 has many more, and that may be a good choice as a scale with which to explore the harmonies of echidna. If for some reason you needed pentads and hexads, 36 notes would be fine.

Triads

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-100-199
 1-9/7-7/6
 swetismic
 4
2
 0-101-199
 1-20/11-7/6
 swetismic
 4
3
 0-100-299
 1-9/7-3/2
 utonal
 6
4
 0-199-299
 1-7/6-3/2
 otonal
 6
5
 0-199-498
 1-7/6-7/4
 utonal
 10
6
 0-299-498
 1-3/2-7/4
 otonal
 10
7
 0-100-596
 1-9/7-9/8
 utonal
 12
8
 0-299-596
 1-3/2-9/8
 ambitonal
 12
9
 0-498-596
 1-7/4-9/8
 otonal
 12
10
 0-101-597
 1-20/11-8/5
 valinorsmic
 12
11
 0-498-597
 1-7/4-8/5
 valinorsmic
 12
12
 0-100-696
 1-9/7-16/11
 pentacircle
 14
13
 0-101-696
 1-20/11-16/11
 otonal
 14
14
 0-596-696
 1-9/8-16/11
 pentacircle
 14
15
 0-597-696
 1-8/5-16/11
 utonal
 14
16
 0-299-894
 1-3/2-6/5
 utonal
 18
17
 0-597-894
 1-8/5-6/5
 otonal
 18
18
 0-101-993
 1-20/11-12/11
 otonal
 20
19
 0-299-993
 1-3/2-12/11
 utonal
 20
20
 0-696-993
 1-16/11-12/11
 otonal
 20
21
 0-894-993
 1-6/5-12/11
 utonal
 20
22
 0-100-1093
 1-9/7-7/5
 swetismic
 22
23
 0-199-1093
 1-7/6-7/5
 utonal
 22
24
 0-498-1093
 1-7/4-7/5
 utonal
 22
25
 0-597-1093
 1-8/5-7/5
 otonal
 22
26
 0-894-1093
 1-6/5-7/5
 otonal
 22
27
 0-993-1093
 1-12/11-7/5
 swetismic
 22
28
 0-101-1192
 1-20/11-14/11
 otonal
 24
29
 0-199-1192
 1-7/6-14/11
 utonal
 24
30
 0-498-1192
 1-7/4-14/11
 utonal
 24
31
 0-596-1192
 1-9/8-14/11
 pentacircle
 24
32
 0-597-1192
 1-8/5-14/11
 valinorsmic
 24
33
 0-696-1192
 1-16/11-14/11
 otonal
 24
34
 0-993-1192
 1-12/11-14/11
 otonal
 24
35
 0-1093-1192
 1-7/5-14/11
 utonal
 24
36
 0-100-1193
 1-9/7-9/5
 utonal
 24
37
 0-299-1193
 1-3/2-9/5
 utonal
 24
38
 0-596-1193
 1-9/8-9/5
 utonal
 24
39
 0-597-1193
 1-8/5-9/5
 otonal
 24
40
 0-894-1193
 1-6/5-9/5
 otonal
 24
41
 0-1093-1193
 1-7/5-9/5
 otonal
 24
42
 0-100-1292
 1-9/7-18/11
 utonal
 26
43
 0-101-1292
 1-20/11-18/11
 otonal
 26
44
 0-199-1292
 1-7/6-18/11
 swetismic
 26
45
 0-299-1292
 1-3/2-18/11
 utonal
 26
46
 0-596-1292
 1-9/8-18/11
 utonal
 26
47
 0-696-1292
 1-16/11-18/11
 otonal
 26
48
 0-993-1292
 1-12/11-18/11
 otonal
 26
49
 0-1093-1292
 1-7/5-18/11
 swetismic
 26
50
 0-1192-1292
 1-14/11-18/11
 otonal
 26
51
 0-1193-1292
 1-9/5-18/11
 utonal
 26

Tetlist

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-100-199-299
 1-9/7-7/6-3/2
 swetismic
 6
2
 0-199-299-498
 1-7/6-3/2-7/4
 ambitonal
 10
3
 0-100-299-596
 1-9/7-3/2-9/8
 utonal
 12
4
 0-299-498-596
 1-3/2-7/4-9/8
 otonal
 12
5
 0-100-596-696
 1-9/7-9/8-16/11
 pentacircle
 14
6
 0-101-597-696
 1-20/11-8/5-16/11
 valinorsmic
 14
7
 0-101-696-993
 1-20/11-16/11-12/11
 otonal
 20
8
 0-299-894-993
 1-3/2-6/5-12/11
 utonal
 20
9
 0-100-199-1093
 1-9/7-7/6-7/5
 swetismic
 22
10
 0-199-498-1093
 1-7/6-7/4-7/5
 utonal
 22
11
 0-498-597-1093
 1-7/4-8/5-7/5
 valinorsmic
 22
12
 0-597-894-1093
 1-8/5-6/5-7/5
 otonal
 22
13
 0-894-993-1093
 1-6/5-12/11-7/5
 swetismic
 22
14
 0-101-199-1192
 1-20/11-7/6-14/11
 swetismic
 24
15
 0-199-498-1192
 1-7/6-7/4-14/11
 utonal
 24
16
 0-498-596-1192
 1-7/4-9/8-14/11
 pentacircle
 24
17
 0-101-597-1192
 1-20/11-8/5-14/11
 valinorsmic
 24
18
 0-498-597-1192
 1-7/4-8/5-14/11
 valinorsmic
 24
19
 0-101-696-1192
 1-20/11-16/11-14/11
 otonal
 24
20
 0-596-696-1192
 1-9/8-16/11-14/11
 pentacircle
 24
21
 0-597-696-1192
 1-8/5-16/11-14/11
 valinorsmic
 24
22
 0-101-993-1192
 1-20/11-12/11-14/11
 otonal
 24
23
 0-696-993-1192
 1-16/11-12/11-14/11
 otonal
 24
24
 0-199-1093-1192
 1-7/6-7/5-14/11
 utonal
 24
25
 0-498-1093-1192
 1-7/4-7/5-14/11
 utonal
 24
26
 0-597-1093-1192
 1-8/5-7/5-14/11
 valinorsmic
 24
27
 0-993-1093-1192
 1-12/11-7/5-14/11
 swetismic
 24
28
 0-100-299-1193
 1-9/7-3/2-9/5
 utonal
 24
29
 0-100-596-1193
 1-9/7-9/8-9/5
 utonal
 24
30
 0-299-596-1193
 1-3/2-9/8-9/5
 utonal
 24
31
 0-299-894-1193
 1-3/2-6/5-9/5
 ambitonal
 24
32
 0-597-894-1193
 1-8/5-6/5-9/5
 otonal
 24
33
 0-100-1093-1193
 1-9/7-7/5-9/5
 swetismic
 24
34
 0-597-1093-1193
 1-8/5-7/5-9/5
 otonal
 24
35
 0-894-1093-1193
 1-6/5-7/5-9/5
 otonal
 24
36
 0-100-199-1292
 1-9/7-7/6-18/11
 swetismic
 26
37
 0-101-199-1292
 1-20/11-7/6-18/11
 swetismic
 26
38
 0-100-299-1292
 1-9/7-3/2-18/11
 utonal
 26
39
 0-199-299-1292
 1-7/6-3/2-18/11
 swetismic
 26
40
 0-100-596-1292
 1-9/7-9/8-18/11
 utonal
 26
41
 0-299-596-1292
 1-3/2-9/8-18/11
 utonal
 26
42
 0-100-696-1292
 1-9/7-16/11-18/11
 pentacircle
 26
43
 0-101-696-1292
 1-20/11-16/11-18/11
 otonal
 26
44
 0-596-696-1292
 1-9/8-16/11-18/11
 pentacircle
 26
45
 0-101-993-1292
 1-20/11-12/11-18/11
 otonal
 26
46
 0-299-993-1292
 1-3/2-12/11-18/11
 ambitonal
 26
47
 0-696-993-1292
 1-16/11-12/11-18/11
 otonal
 26
48
 0-100-1093-1292
 1-9/7-7/5-18/11
 swetismic
 26
49
 0-199-1093-1292
 1-7/6-7/5-18/11
 swetismic
 26
50
 0-993-1093-1292
 1-12/11-7/5-18/11
 swetismic
 26
51
 0-101-1192-1292
 1-20/11-14/11-18/11
 otonal
 26
52
 0-199-1192-1292
 1-7/6-14/11-18/11
 swetismic
 26
53
 0-596-1192-1292
 1-9/8-14/11-18/11
 pentacircle
 26
54
 0-696-1192-1292
 1-16/11-14/11-18/11
 otonal
 26
55
 0-993-1192-1292
 1-12/11-14/11-18/11
 otonal
 26
56
 0-1093-1192-1292
 1-7/5-14/11-18/11
 swetismic
 26
57
 0-100-1193-1292
 1-9/7-9/5-18/11
 utonal
 26
58
 0-299-1193-1292
 1-3/2-9/5-18/11
 utonal
 26
59
 0-596-1193-1292
 1-9/8-9/5-18/11
 utonal
 26
60
 0-1093-1193-1292
 1-7/5-9/5-18/11
 swetismic
 26

Pentlist

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-101-597-696-1192
 1-20/11-8/5-16/11-14/11
 valinorsmic
 24
2
 0-101-696-993-1192
 1-20/11-16/11-12/11-14/11
 otonal
 24
3
 0-199-498-1093-1192
 1-7/6-7/4-7/5-14/11
 utonal
 24
4
 0-498-597-1093-1192
 1-7/4-8/5-7/5-14/11
 valinorsmic
 24
5
 0-100-299-596-1193
 1-9/7-3/2-9/8-9/5
 utonal
 24
6
 0-597-894-1093-1193
 1-8/5-6/5-7/5-9/5
 otonal
 24
7
 0-100-199-299-1292
 1-9/7-7/6-3/2-18/11
 swetismic
 26
8
 0-100-299-596-1292
 1-9/7-3/2-9/8-18/11
 utonal
 26
9
 0-100-596-696-1292
 1-9/7-9/8-16/11-18/11
 pentacircle
 26
10
 0-101-696-993-1292
 1-20/11-16/11-12/11-18/11
 otonal
 26
11
 0-100-199-1093-1292
 1-9/7-7/6-7/5-18/11
 swetismic
 26
12
 0-101-199-1192-1292
 1-20/11-7/6-14/11-18/11
 swetismic
 26
13
 0-101-696-1192-1292
 1-20/11-16/11-14/11-18/11
 otonal
 26
14
 0-596-696-1192-1292
 1-9/8-16/11-14/11-18/11
 pentacircle
 26
15
 0-101-993-1192-1292
 1-20/11-12/11-14/11-18/11
 otonal
 26
16
 0-696-993-1192-1292
 1-16/11-12/11-14/11-18/11
 otonal
 26
17
 0-199-1093-1192-1292
 1-7/6-7/5-14/11-18/11
 swetismic
 26
18
 0-993-1093-1192-1292
 1-12/11-7/5-14/11-18/11
 swetismic
 26
19
 0-100-299-1193-1292
 1-9/7-3/2-9/5-18/11
 utonal
 26
20
 0-100-596-1193-1292
 1-9/7-9/8-9/5-18/11
 utonal
 26
21
 0-299-596-1193-1292
 1-3/2-9/8-9/5-18/11
 utonal
 26
22
 0-100-1093-1193-1292
 1-9/7-7/5-9/5-18/11
 swetismic
 26

Hexad

Number
 Chord
 Transversal
 Type
 Complexity
1
 0-101-696-993-1192-1292
 1-20/11-16/11-12/11-14/11-18/11
 otonal
 26
2
 0-100-299-596-1193-1292
 1-9/7-3/2-9/8-9/5-18/11
 utonal
 26